DSpace at VNU: An existence theorem for generalized variational inequalities with discontinuous and pseudomonotone operators

Nonlinear Analysis 74 (2011) 1495–1500 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na An existence theorem for generalized variational inequalities with discontinuous and pseudomonotone operators✩ B.T Kien a , G.M Lee b,∗ a Department of Information and Technology, Hanoi National University of Civil Engineering, 55 Giai Phong, Hanoi, Viet Nam b Department of Applied Mathematics, Pukyong National University, Busan, 608-737, Republic of Korea article info Article history: Received 10 June 2010 Accepted 11 October 2010 Keywords: B-pseudomonotone operator K-pseudomonotone operator C-pseudomonotone operator Generalized variational inequality Solution existence abstract This paper gives a solution existence theorem for a generalized variational inequality problem with an operator which is defined on an infinite dimensional space, which is C-pseudomonotone in the sense of Inoan and Kolumbán [D Inoan, J Kolumbán, On pseudomonotone set-valued mappings, Nonlinear Analysis 68 (2008) 47–53], but which may not be upper semicontinuous on finite dimensional subspaces The proof of the theorem provides a new technique which reduces infinite variational inequality problems to finite ones Two examples are given and analyzed to illustrate the theorem Moreover, an example is presented to show that the C-pseudomonotonicity of the operator cannot be omitted in the theorem © 2010 Elsevier Ltd All rights reserved Introduction The theory of pseudomonotone operators plays an important role in nonlinear analysis, optimization and variational inequalities (see [1–10]) There are some kinds of pseudomonotone operators which were studied One type of pseudomonotone operators was introduced by Karamardian in 1976 [6] which has been frequently used in optimization problems (see [2,4,11]) Another type of pseudo-monotone operators was introduced by Brezis in 1968 [12] which has been used in the study of solution existence of partial differential equations and integral equations (see [13,14]) It is known that the two mentioned classes of pseudo-monotone operators are different However, recently, it has been shown by [3,5] that both notions have a common generalization which is useful for studying generalized variational inequalities Namely, under certain conditions, both classes satisfy a property of the so-called C -pseudomonotone operators Let us recall some notations and concepts which are related to our problem Throughout the paper we assume that X is a topological vector space which satisfies the Hausdoff separation axiom and X ∗ is its dual Suppose K ⊂ X is a nonempty ∗ closed convex set and Φ : K → 2X is a multifunction from K to X ∗ (which is equipped with the weak∗ topology) We denote by Gr(Φ ) the graph of Φ , which is defined by Gr(Φ ) = {(x, x∗ ) ∈ K × X ∗ : x∗ ∈ Φ (x)} The generalized variational inequality defined by Φ and K , denoted by GVI(Φ , K ), is the problem of finding a point x ∈ K and x∗ ∈ Φ (x) such that ⟨x∗ , y − x⟩ ≥ ∀y ∈ K (1) ✩ This work was supported by the Korea Science and Engineering Foundation (KOSEF) NRL Program grant funded by the Korea government (MEST) (No ROA-2008-000-20010-0) ∗ Corresponding author E-mail addresses: kienbt@nuce.edu.vn (B.T Kien), gmlee@pknu.ac.kr (G.M Lee) 0362-546X/$ – see front matter © 2010 Elsevier Ltd All rights reserved doi:10.1016/j.na.2010.10.022 1496 B.T Kien, G.M Lee / Nonlinear Analysis 74 (2011) 1495–1500 Recall that Φ is said to be monotone if for all (x, x∗ ), (y, y∗ ) ∈ Gr(Φ ) one has ⟨x∗ − y∗ , x − y⟩ ≥ Φ is said to be pseudomonotone in the sense of Karamardian (K -pseudomonotone, for short) if for any (x, x∗ ), (y, y∗ ) ∈ Gr(Φ ) the following implication holds: ⟨y∗ , x − y⟩ ≥ ⇒ ⟨x∗ , x − y⟩ ≥ It is clear that monotonicity implies K -pseudomonotonicity ∗ Φ : K → 2X is called pseudomonotone in the sense of Brezis (B-pseudomonotone, for short) if for any u ∈ K and every net {ui } with ui ⇀ u, u∗i ∈ Φ (ui ), and lim sup⟨u∗i , ui − u⟩ ≤ then for each v ∈ K , there exists u∗v ∈ Φ (u) such that ⟨u∗v , u − v⟩ ≤ lim inf⟨u∗i , ui − v⟩ It is known that if Φ is monotone and hemicontinuous then Φ is B-pseudomonotone (see [15, Proposition 27.6]) Moreover, when X is a finite dimensional space, any continuous single-valued map φ : K → X ∗ is B-pseudomonotone But it is not necessary to be K -pseudomonotone ∗ According to [5], Φ : K → 2X is said to be C -pseudomonotone if for any x, y ∈ K and net {xi } in K with xi ⇀ x, sup ⟨x∗ , (1 − t )x + ty − xi ⟩ ≥ 0, x∗ ∈Φ (xi ) ∀t ∈ [0, 1], ∀i ∈ I implies sup ⟨x∗ , y − x⟩ ≥ x∗ ∈Φ (x) The C -pseudomonotone maps appeared in the literature also under the name of 0-segmentary closed maps, see for instance [16] The implications ‘‘B-pseudomonotone implies C -pseudomonotone’’ and ‘‘K -pseudomonotone and hemicontinuous implies C -pseudomonotone’’ are proved in [5] in the presence of some supplementary conditions Based on a characteristic property of C -pseudomonotone operators and using a generalization of the Fan intersection lemma, Inoan and Kolumbán [5] established an important existence theorem for generalized variational inequalities Observe that, in almost all previous results and the result of [5] on the solution existence of GVIs, the continuity of operator Φ is often required Namely, solution existence of GVIs is guaranteed under assumptions that Φ is monotone or pseudomonotone (in some sense) and Φ is upper semicontinuous on finite dimensional subspaces of X (see for instance [7, Chapter III Theorem 1.4] and [5, Theorem 15]) One may ask whether the solution existence of GVIs for C -pseudomonotone operators is still valid if operators are not continuous on finite dimensional subspaces The aim of this paper is to answer the above question We will show that if Φ is C -pseudomonotone and satisfies some additional conditions, the solution existence of GVI(Φ , K ) is still guaranteed even though Φ is discontinuous Notice that, there are some papers dealing with solution existence of variational inequalities with discontinuous data in the literature (see for instance [17,18]) But GVIs were considered in the finite dimensional setting, and the obtained results were proved and followed directly from the so-called KKM lemma In this paper the problem is considered in infinite dimensional spaces Besides, it seems that our main result cannot follow directly from the Fan intersection lemma [19] The obtained theorem is proved by a technique which reduce infinite problems to finite problems Although the obtained result is modest, the contribution here is to give a new method of proof We are now ready for stating our result ∗ Theorem 1.1 Let X be a topological vector space and K ⊂ X be convex, closed and nonempty Suppose that Φ : K → 2X is an operator which satisfies the following conditions: (i) Φ is C -pseudomonotone, (ii) there exist weakly compact subsets B0 , B of K , where B0 ⊆ B and B0 lies in a finite dimensional subspace, such that for every x ∈ K \ B there exists z ∈ B0 satisfying sup ⟨f , z − x⟩ < 0, f ∈Φ (x) (iii) for each z ∈ K − K , the set {x ∈ K : supx∗ ∈Φ (x) ⟨x∗ , z ⟩ ≥ 0} is closed, (iv) Φ has convex and weakly∗ compact values Then GVI(Φ , K ) has a solution, that is, there exist x0 ∈ B and x∗0 ∈ Φ (x0 ) such that ⟨x∗0 , x − x0 ⟩ ≥ 0, ∀x ∈ K B.T Kien, G.M Lee / Nonlinear Analysis 74 (2011) 1495–1500 1497 Proof of Theorem 1.1 Let us denote by L the family of finite dimensional subspaces L ⊂ X such that B0 ⊂ L For each L ∈ L, we set BL = B ∩ L and KL = K ∩ L It is clear that BL and KL are closed Fixing any L ∈ L, we define a mapping ΦL : L ∩ K → L∗ by the formula ΦL (x) = {αL x∗ : x∗ ∈ Φ (x)}, where αL : X ∗ → L∗ is given by (2) ⟨αL x∗ , y⟩ = ⟨x∗ , y⟩ ∀y ∈ L (3) To prove the theorem we use the following lemma n Lemma 2.1 ([18, Theorem 1.2]) Suppose K ⊂ Rn is a nonempty closed convex set and T : K → 2R is a multifunction which satisfies the following two conditions: (a) for each z ∈ K − K , the set {x ∈ K : supf ∈T (x) ⟨f , z ⟩ ≥ 0} is closed, (b) there exists a compact subset B ⊂ K with the property that for every x ∈ K \ B, there exists z ∈ B such that sup ⟨f , z − x⟩ < 0, f ∈T (x) (c) T has convex and compact values Then there exist x0 ∈ B and f0 ∈ T (x0 ) such that ⟨f0 , x − x0 ⟩ ≥ 0, ∀x ∈ K Consider the problem GVI(ΦL , KL ) We shall show that GVI(ΦL , KL ) satisfies all conditions of Lemma 2.1 (a) For each z ∈ KL − KL , z ∈ L and so by (2) we have {x ∈ KL : sup ⟨f , z ⟩ ≥ 0} = {x ∈ KL : sup ⟨αL x∗ , z ⟩ ≥ 0} f ∈ΦL (x) x∗ ∈Φ (x) = {x ∈ KL : sup ⟨x∗ , z ⟩ ≥ 0} = {x ∈ K : sup ⟨x∗ , z ⟩ ≥ 0} ∩ L, x∗ ∈Φ (x) x∗ ∈Φ (x) which is a closed set by condition (iii) of Theorem 1.1 (b) Since KL \ BL ⊂ K \ B, it follows from condition (ii) of Theorem 1.1 that for each x ∈ KL \ BL there exists z ∈ B0 ⊂ BL such that sup ⟨f , z − x⟩ = sup ⟨x∗ , z − x⟩ < f ∈ΦL (x) x∗ ∈Φ (x) (c) Since Φ has convex and weakly∗ compact values, ΦL has convex and compact values on KL Thus all conditions of Lemma 2.1 are fulfilled According to Lemma 2.1, for each L ∈ L, there exists xL ∈ BL which is a solution of GVI(ΦL , KL ), that is, sup ⟨f , y − xL ⟩ ≥ ∀y ∈ KL f ∈ΦL (xL ) This is equivalent to sup ⟨x∗ , y − xL ⟩ ≥ ∀y ∈ KL x∗ ∈Φ (xL ) (4) For each Y ∈ L we denote by SY the set of all xˆ ∈ B such that there exists a subspace L ⊇ Y with the property that xˆ ∈ BL and sup ⟨x∗ , y − xˆ ⟩ ≥ x∗ ∈Φ (ˆx) ∀ y ∈ KL We claim that the family {S Y } has the finite intersection property, where S Y is the weak closure of SY In fact, for each Y ∈ L, by putting L = Y , we have from (4) that xY ∈ SY Hence S Y is nonempty Take subspaces L1 , L2 , , Ln ∈ L and put M = span{L1 , L2 , , Ln } Then we have M ∈ L and SM ⊂ n  SLi i =1 This implies that ∅ ̸= SM ⊆ S M ⊆ n  i =1 SLi ⊆ n  S Li i =1 The claim is proved Since S Y ⊂ B and B is weakly compact, the finite intersection property of {S Y } implies  Y ∈L S Y ̸= ∅ 1498 B.T Kien, G.M Lee / Nonlinear Analysis 74 (2011) 1495–1500 This means that there exists a point x0 ∈ B such that x0 ∈ S Y for all Y ∈ L Fix any y ∈ K and choose Y ∈ L such that Y contains y and x0 Since x0 ∈ S Y , there exists a net xi ∈ SY such that xi ⇀ x0 By definition of SY we have sup ⟨x∗ , v − xi ⟩ ≥ ∀ v ∈ KY x∗ ∈Φ (xi ) In particular, for v = ty + (1 − t )x0 , we get sup ⟨x∗ , ty + (1 − t )x0 − xi ⟩ ≥ ∀t ∈ [0, 1], ∀i ∈ I x∗ ∈Φ (xi ) (5) Since Φ is C -pseudomonotone, from (5) we get sup ⟨x∗ , y − x0 ⟩ ≥ (6) x∗ ∈Φ (x0 ) To obtain the conclusion we need the following Sion’s minimax theorem (see also [20, Theorem 1]) Lemma 2.2 ([21, Theorem 3.4]) Let P be a compact convex set in a topological vector space X and Q be a convex subset of a topological vector space Y Let h be a real-valued function on P × Q such that (i) h(x, ·) is upper semicontinuous and quasi-concave on Q for each x ∈ P, (ii) h(·, y) is lower semicontinuous and quasi-convex on P for each y ∈ Q Then sup h(x, y) = sup h(x, y) x∈P y∈Q y∈Q x∈P Since y is arbitrary, from (6) we have sup ⟨x∗ , x − x0 ⟩ ≥ 0, x∗ ∈Φ (x0 ) ∀x ∈ K This is equivalent to ⟨x∗ , x0 − x⟩ ≤ 0, x∗ ∈Φ (x0 ) ∀x ∈ K Hence sup ⟨x∗ , x0 − x⟩ ≤ ∗ x∈K x ∈Φ (x0 ) By Lemma 2.2, we get sup⟨x∗ , x0 − x⟩ = sup ⟨x∗ , x0 − x⟩ ≤ x∗ ∈Φ (x0 ) x∈K ∗ x∈K x ∈Φ (x0 ) (7) Since the function φ(x∗ ) := supx∈K ⟨x∗ , x0 − x⟩ is lower semicontinuous in the weakly star tolology of X ∗ , there exist x∗0 ∈ Φ (x0 ) such that φ(x∗0 ) = minx∗ ∈Φ (x0 ) φ(x∗ ) Hence from (7) we obtain ⟨x∗0 , x0 − x⟩ ≤ 0, ∀x ∈ K The proof of the theorem is complete A special case and examples Let us give a special case of Theorem 1.1 when Φ is upper semicontinuous Recall that a multifunction G : K ⊂ X → 2Y , where Y is a topological space, is said to be upper semicontinuous on K if for any closed set V ⊂ Y , the set {x : G(x) ∩ V ̸= ∅} is closed Corollary 3.1 (Cf [5, Theorem 15]) Let X be a topological vector space and K ⊂ X be convex, closed, nonempty Suppose that ∗ Φ : K → 2X is an operator which satisfies the following conditions (a) Φ is C -pseudomonotone, (b) there exist a weakly compact subset B ⊂ X and z0 ∈ B such that sup ⟨f , z0 − x⟩ < f ∈Φ (x) for all x ∈ K \ B, (c) for every finite dimensional subspace Z of X , Φ is upper semicontinuous on K ∩ Z , with the weak∗ topology in X ∗ , (d) Φ (x) is convex and weakly∗ compact for every x ∈ K Then GVI(Φ , K ) has a solution B.T Kien, G.M Lee / Nonlinear Analysis 74 (2011) 1495–1500 1499 Proof For the proof we put B0 = {z0 } and show that for each L ∈ L, GVI(ΦL , KL ) has a solution xL ∈ BL = B ∩ L To this we verify all conditions of Lemma 2.1 It is clear that conditions (b) and (c) of Lemma 2.1 are automatically fulfilled for GVI(ΦL , KL ) It remains to check condition (a) For each z ∈ KL − KL , we have {x ∈ K : sup ⟨x∗ , z ⟩ ≥ 0} ∩ L = {x ∈ KL : sup ⟨f , z ⟩ ≥ 0} f ∈ΦL (x) x∗ ∈Φ (x) = {x ∈ KL : ΦL (x) ∩ W ̸= ∅}, where W = {f ∈ L : ⟨f , z ⟩ ≥ 0}, which is a closed set in L∗ By (c), ΦL is upper semicontinuous on KL , it follows that the set {x ∈ KL : ΦL (x) ∩ W ̸= ∅} is closed Hence (a) of Lemma 2.1 is valid The conclusion follows from the proof method of Theorem 1.1 ∗ Remark 3.2 The conditions of [5, Theorem 15] are the same as Corollary 3.1 except for the following: (b)′ there exist a weakly compact subset B ⊂ X and z0 ∈ K such that sup ⟨f , z0 − x⟩ < ∀x ∈ K \ B; f ∈Φ (x) (d)′ Φ (x) is weakly∗ compact for every x ∈ K The following are some illustrative examples for Theorem 1.1, where solution existence of GVI(Φ , K ) is guaranteed even though Φ is discontinuous Example 3.3 Suppose X = R, K = [0, 1] and φ : K → R is defined by φ(0) = 0, φ(x) = for all x ̸= Then φ is C -pseudomonotone and discontinuous at For each z ∈ K − K we have {x ∈ K : ⟨φ(x), z ⟩ ≥ 0} =  {0} [0, 1] if z < if z ≥ Hence all conditions of Theorem 1.1 are fulfilled But Corollary 3.1 or Theorem 15 in [5] cannot apply to this example By a simple computation, we see that VI(φ, K ) has a solution x0 = Let us give an example for the case of infinite dimensional spaces Example 3.4 Let  X = l2 = x = (x1 , x2 , , xn , ) : ∞ −  |xi | < +∞ i=1 and K = {x = (x1 , x2 , , xk , ) : ‖x‖l2 ≤ 1} Then K is a convex and weakly compact set in l2 Let Φ : K → l2 be an operator which is defined by Φ (x) =  {θ} {y0 = (1, 0, 0, , 0, )} if x = θ if x ̸= θ , where θ = (0, 0, , 0, ) We would like to check the conditions of Theorem 1.1 It is obvious that conditions (ii) and (iv) are automatically fulfilled Taking any z ∈ K − K , we assume that z = (z1 , z2 , , zn , ) Then we have {x ∈ K : sup ⟨x∗ , z ⟩ ≥ 0} = x∗ ∈Φ (x)  if z1 < if z1 ≥ {θ} K Hence condition (iii) of Theorem 1.1 is valid We now claim that Φ is C -pseudomonotone on K Let x, y ∈ K , xj ∈ K and xj ⇀ x such that sup ⟨f , z − xj ⟩ ≥ 0, f ∈Φ (xj ) ∀z ∈ [x, y], ∀j ∈ N j j j Let x = (x1 , x2 , , xk , ) and xj = (x1 , x2 , , xk , ) If x = θ then sup ⟨f , y − θ ⟩ ≥ f ∈Φ (θ) 1500 B.T Kien, G.M Lee / Nonlinear Analysis 74 (2011) 1495–1500 If x ̸= θ then there exists j0 such that xj ̸= θ for all j ≥ j0 , and so we have sup ⟨f , z − xj ⟩ = ⟨y0 , z − xj ⟩ ≥ 0, f ∈Φ (xj ) ∀z ∈ [x, y], ∀j ≥ j0 (8) j Note that since xj ⇀ x, we have xk → xk for all k = 1, 2, Putting z = y and letting j → ∞, we get from (8) that sup ⟨f , y − x⟩ = ⟨y0 , y − x⟩ ≥ f ∈Φ (x) Hence Φ is C -pseudomonotone and so condition (i) of Theorem 1.1 is valid Thus all conditions of Theorem 1.1 are fulfilled Observe that for any z ∈ K we have sup ⟨f , z − θ ⟩ = f ∈Φ (θ) Hence θ is a solution of VI(Φ , K ) Besides, Φ is discontinuous at θ From Example 3.4 one may ask whether GVI(Φ , K ) still has a solution if the assumption on C-pseudomonotonicity of Φ is omitted The following example shows that condition (i) of Theorem 1.1 plays an essential role for the solution existence of GVI(Φ , K ) Example 3.5 Suppose X = l2 and K is the unit ball in X Let Φ : K → l2 defined by Φ (x) = ( − ‖x‖2 , x1 , x2 , , xk , ) − x with x = (x1 , x2 , , xk , ) It is clear that Φ is continuous Hence condition (iii) is valid Besides, conditions (ii) and (iv) are automatically fulfilled However, Φ is not C -pseudomonotone on K In fact, taking x = θ = (0, 0, , 0, ), y = (−1, 0, 0, , 0, ) and a sequence xj = (0, 0, , 0, 1, 0, 0, ), where is at the j-th position, we see that xj converges weakly to x and for all t ∈ [0, 1] one has  ⟨Φ (xj ), ty + (1 − t )x − xj ⟩ =  t +1 if j = 1, if j > But we have ⟨Φ (x), y − x⟩ = −1 < Thus Φ is not C -pseudomonotone on K We now show that GVI(Φ , K )has no solution Conversely, suppose x is a solution of the problem If ‖x‖ < then we get Φ (x) = This implies that ( − ‖x‖2 , x1 , x2 , , xk , ) = x Hence ‖x‖ = 1, which is absurd If ‖x‖ = then the condition ⟨Φ (x), y − x⟩ ≥ ∀y ∈ K , yields Φ (x) = −λx for some λ ≥ This implies = |1 − λ| and so λ = or λ = When λ = then we get ( − ‖x‖2 ,   x1 , x2 , , xk , ) = x From this we obtain x = θ Also, when λ = we get ( − ‖x‖2 , x1 , x2 , , xk , ) = −x It follows that x = θ , which contradicts ‖x‖ = Therefore GVI(Φ , K ) has no solution Acknowledgements The authors wish to thank the anonymous referees for their suggestions and comments 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